# Circuits 2

**Main Reference Book**

**Main Reference Book**

Tim Hanshaw, “** Real Analog: An Introduction to Electrical Circuits**”, free online access

**Additional Reference**

**Additional Reference**

Roland E. Thomas, Albert J. Rosa, and Gregory J. Toussaint

“** The Analysis and Design of Linear Circuits**”, Wiley, 7th Edition, 2012

(Prior or more recent versions of this book are also acceptable)

**Course Goals**

- AC analysis of linear circuits to include circuit theorems via classical and transform techniques.
- Emphasis is placed on the Laplace transform, including use of pole-zero and Bode diagrams to analyze and design circuits, including multiple filters (single pole cascade, Butterworth, Chebychev), and step response circuits.
- Phasor applications to sinusoidal steady state analysis and AC power.
- Computer analysis software is used as an aid to circuit analysis.
- Laboratory program practicing time and frequency domain analysis and design techniques on step response and filter problems.
- Applications to instrumentation and circuits.

**Course Modules**

**Module 0 – Energy Storage Elements (Review)**

- Qualitatively state the effect of energy storage on the type of mathematics governing a system
- Define transient response
- Define steady-state response
- Write the mathematical expression for a unit step function
- Sketch the unit step function
- Sketch shifted and scaled versions of the unit step function
- Write the mathematical expression for a decaying exponential function
- Define the time constant of an exponential function
- Sketch a decaying exponential function, given the function’s initial value and time constant
- Use a unit step function to restrict an exponential function to times greater than zero
- Write the circuit symbol for a capacitor
- State the mechanism by which a capacitor stores energy
- State the voltage-current relationship for a capacitor in both differential and integral form
- State the response of a capacitor to constant voltages and instantaneous voltage changes
- Write the mathematical expression describing energy storage in a capacitor
- Determine the equivalent capacitance of series and parallel combinations of capacitors
- Sketch a circuit describing a non-ideal capacitor
- Write the circuit symbol for an inductor
- State the mechanism by which an inductor stores energy
- State the voltage-current relationship for an inductor in both differential and integral form
- State the response of an inductor to constant voltages and instantaneous current changes
- Write the mathematical expression describing energy storage in an inductor
- Determine the equivalent inductance of series and parallel combinations of inductors
- Sketch a circuit describing a non-ideal inductor

**Module 1 – First Order Circuits**

- Write the general form of the differential equation governing a first order system
- State, in physical terms, the significance of a differential equation’s homogeneous and particular solutions
- Define, from memory, the relationships between a system’s unforced response, zero-input response, natural response, and the homogeneous solution to the differential equation governing the system
- Define, from memory, the relationships between a system’s forced response, zero-state response, and the particular solution to the differential equation governing the system
- Determine the time constant of a first order system from the differential equation governing the system
- Write mathematical expressions from memory, giving the form of the natural and step responses of a first order system
- Sketch the natural response of a first order system from the differential equation governing the system and the system’s initial condition
- Sketch the step response of a first order system from the differential equation governing the system and the amplitude of the input step function
- Write the differential equation governing RC and RL circuits
- Determine the time constant of RC and RL circuits from their governing differential equations
- Determine the time constant of RC and RL circuits directly from the circuits themselves
- Determine initial conditions on arbitrary RC and RL circuits
- Write from memory the form of the natural responses of RC and RL circuits
- Determine the natural response of RC and RL circuits, given the governing differential equation and initial conditions
- Write the form of the differential equations governing forced first order electrical circuits
- Determine the time constant of a forced electrical circuit from the governing differential equation
- Write differential equations governing passive and active first order circuits
- Determine the differential equation governing the step response of a first order electrical circuit
- Write the form of the particular solution of a first order differential equation, to a step input
- Write the form of the step response of a first order electrical circuit
- Determine the final conditions (steady-state response) of a first order electrical circuit, to a step input
- Define DC gain for a circuit and relate it to the steady-state response to a step input
- Determine the step response of a first order electrical circuit from the governing differential equation, the initial conditions, and the final conditions

**Module 2 – Second Order Circuits**

- Write differential equations governing second order circuits
- Define damping ratio and natural frequency from the coefficients of a second order differential equation
- Express the form of the natural response of an arbitrary second order system in terms of complex exponentials, the damping ratio, and the natural frequency
- Summarize the behavior of the complex exponentials in the system natural response for the damping ratio ranges below:
- Damping ratio greater than one
- Damping ratio less than one
- Damping ratio equal to one

- Write complex numbers in terms of complex exponentials
- Express sinusoidal signals in terms of complex exponentials
- Classify
*overdamped*,*underdamped*, and*critically damped*systems according to their damping ratio - Identify the expected shape of the natural response of over-, under-, and critically damped systems
- State from memory the definition of an underdamped second order system’s overshoot, rise time, and steady-state response
- Use the coefficients of a second order system’s governing equation to estimate the system’s overshoot, rise time, and steady-state response

**Module 3 – Laplace (s-Domain)**

- Video 1 – Introduction
- Video 2 – Flow Diagram
- Video 3 – s-Domain Models
- Video 4 – Algebraic Equation
- Video 5 – Initial and Final Value Theorem
- Video 6 – Partial Fraction Expansion 1
- Video 7 – Inverse Laplace
- Video 8 – Plot
- Video 9 – Partial Fraction Expansion 2
- Video 10 – Practice Example

**Videos**

**Module 4 – Introduction to State Variable Models**

- Define state variables for electrical circuits
- Write differential equations governing electrical circuits in state variable form
- Use MATLAB and/or Octave to simulate the impulse response of an electrical circuit
- Use MATLAB and/or Octave to simulate the step response of an electrical circuit
- Use MATLAB and/or Octave to plot the state trajectory of an electrical circuit

**Module 5 – Steady-State Analysis – Phasors**

- State the relationship between the sinusoidal steady state system response and the forced response of a system
- For sinusoidal steady-state conditions, state the relationship between the frequencies of the input and output signals for a linear, time-invariant system
- State the two parameters used to characterize the sinusoidal steady-state response of a linear, time invariant system
- Express sinusoidal signals in phasor form
- Perform frequency-domain analyses of electrical circuits
- Sketch phasor diagrams of a circuit’s input and output
- State the definition of impedance and admittance
- State how to use the following analysis approaches in the frequency domain:
- KVL and KCL
- Voltage and current dividers
- Circuit reduction techniques
- Nodal and mesh analysis
- Superposition, especially when multiple frequencies are present
- Thévenin’s and Norton’s theorems

**Module 6 – Steady-State Analysis – Fourier**

- Video 1 – Fourier Intro
- Video 2 – Fourier Series vs Fourier Transform
- Video 3 – Fourier Series – Alternative Form
- Video 4 – Fourier Coefficients
- Video 5 – Amplitude and Phase Spectrum
- Video 6 – Circuit Analysis
- Video 7 – Practice Circuit
- Video 8 – Fourier Transform Intro
- Video 9 – Fourier Transform (Path 1 part a)
- Video 10 – Fourier Transform (Path 1 part b)
- Video 11 – Fourier Transform (Path 2)
- Video 12 – Fourier vs Laplace

**Module 7 – Frequency Response and Filtering**

- Use the frequency response of a system to determine the frequency domain response of a system to a given input
- State from memory the definition of
*signal spectrum* - Create plots of given signal spectra
- Plot a circuit’s magnitude and phase responses
- Check a circuit’s amplitude response at low and high frequencies against the expected physical behavior of the circuit
- Graphically represent a system’s frequency domain response from provided signal spectra plots and plots of the system’s frequency response
- Identify low pass and high pass filters
- Calculate a system’s cutoff frequency
- Determine the DC gain of an electrical circuit
- Write, from memory, the equation used to convert gains to decibel form
- Sketch straight-line amplitude approximations to Bode plots
- Sketch straight-line phase approximations to Bode plots

**Module 8 – Steady-State Sinusoidal Power**

- Define instantaneous power, average power, and reactive power
- Define real power, reactive power, and complex power
- Define RMS signal values and calculate the RMS value of a given sinusoidal signal
- State, from memory, the definition of power factor and calculate the power factor from a given combination of voltage and current sinusoids
- Draw a power triangle
- Correct the power factor of an inductive load to a desired value